Spectral SLLN and Pointwise Ergodic Theorem in L alpha.

摘要

The criterion, obtained by Gaposhkin, for a (weakly) stationary process to satisfy the strong law of large numbers (SLLN) has had various extensions, in particular to second order non-stationary harmonizable processes. Outside of the L squared-framework, it has also been studied for Fourier transforms of independently scattered symmetric alpha-stable (S alpha S) measures. It is shown here that via this spectral approach, neither the L squared - requirement nor any distributional assumption are indispensable in establishing the SLLN. Only the harmonic representation with respect to a bounded (in a sense to be made precise) random measure is crucial. This is illustrated in this work, where we obtain conditions for the SLLN to hold for some classes of processes with finite alpha sub th-moment, which, in addition, are Fourier transforms. It is well known that stationary processes and unitary groups of operators are interchangeable, and so are the corresponding strong law and pointwise ergodic theorem. This type of duality between operators and processes carries over to our framework, although in general, the operators are not shifts. It is thus, also the purpose of our work to obtain the pointwise ergodic theorem, for some new classes of operators between L sub alpha-spaces, 1 < alpha < + infinity. Keywords: Ergodic properties, Random fields. (KR)